At which point [in the Hallel service] did they wave the [lulav]? At “Give thanks to HASHEM,” at the beginning and end, and at “Please, HASHEM, bring salvation now”; these are the words of Beis Hillel. Beis Shammai say: Also at “Please, HASHEM, bring success now.”  Said R’ Akiva: As I was watching Rabban Gamliel and R’ Yehoshua, all the people were waving their lulavim, but they did not wave anything except during “Please, HASHEM, bring salvation now.”  One who arrived from a journey where he did not have a lulav at hand to use, when he comes home he should take it at his table.  If he did not take it in the morning he should take it in the afternoon, for the entire day is valid for the lulav.

If a [non-Jewish] slave, a woman, or a minor recited [Hallel] for someone, he must repeat after them whatever they say —- and let it be a curse upon him!  If an adult was reciting for him, he must respond after him, “Hallelu Yah.”